JEE Mathematics Linear Programming MCQs Set 03

Download JEE MCQs for JEE Mathematics: Linear Programming

Review structured MCQ sets for JEE Mathematics Linear Programming. Built according to official JEE guidelines, these downloadable questions support daily revision and core concept reinforcement.

Chapter-wise Objective Questions: Linear Programming

Access the complete set of multiple-choice questions for Linear Programming below. This focused format allows students to isolate specific topics for thorough review and uninterrupted practice alongside official JEE textbooks.

Multiple Choice Questions

Question 1: An oil company requires \(12000\), \(20000\) and \(15000\) barrels of high-, medium- and low-grade oil. Refinery \(A\) produces \(100\), \(300\), \(200\) barrels per day of high-, medium-, low-grade oil and refinery \(B\) produces \(200\), \(400\), \(100\) barrels per day. Refinery \(A\) costs Rs. \(400\) per day and \(B\) costs Rs. \(300\) per day. The numbers of days \(A\) and \(B\) should each be run to minimise cost while meeting the requirements are
(a) \(30\) days and \(60\) days
(b) \(60\) days and \(30\) days
(c) \(40\) days and \(60\) days
(d) \(60\) days and \(40\) days
Show Answer & Explanation

Answer: (b) \(60\) days and \(30\) days

Explanation:
1. Constraints: \(x+2y\ge120\), \(3x+4y\ge200\), \(2x+y\ge150\), \(x,y\ge0\); minimise \(Z=400x+300y\).
2. Corner points: \((0,150)\): \(45000\); \((60,30)\): \(33000\); \((120,0)\): \(48000\). The minimum is at \((60,30)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 2: A vertex of the bounded region of the inequalities \(x\ge0\), \(x+2y\ge0\) and \(2x+y\le4\) (the region is unbounded below; consider the corner points of its boundary) is
(a) \((1,1)\)
(b) \((0,1)\)
(c) \((0,4)\)
(d) \((3,0)\)
Show Answer & Explanation

Answer: (c) \((0,4)\)

Explanation:
1. The boundary lines meet at \((0,0)\) (\(x=0\) and \(x+2y=0\)), at \((0,4)\) (\(x=0\) and \(2x+y=4\)) and at \(\left(\dfrac83,-\dfrac43\right)\).
2. Among the options only \((0,4)\) is a corner point.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 3: Which of these terms is NOT used in a linear programming problem?
(a) Slack variables
(b) Objective function
(c) Feasible solution
(d) Concave region
Show Answer & Explanation

Answer: (d) Concave region

Explanation:
1. Slack variables, the objective function and feasible solutions are standard LPP terms.
2. The feasible region is convex, so 'concave region' is not used.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 4: The maximum value of \(z=6x+8y\) subject to \(2x+y\le30\), \(x+2y\le24\), \(x\ge0\), \(y\ge0\) is
(a) \(120\)
(b) \(90\)
(c) \(96\)
(d) \(240\)
Show Answer & Explanation

Answer: (a) \(120\)

Explanation:
1. Corner points: \((0,0)\), \((15,0)\), \((12,6)\), \((0,12)\).
2. \(z=0,\ 90,\ 120,\ 96\), so the maximum is \(120\) at \((12,6)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 5: Minimise \(z=3x+2y\) subject to \(x+y\ge8\), \(3x+5y\le15\), \(x,y\ge0\). The problem has
(a) infinite feasible solutions
(b) a unique feasible solution
(c) no feasible solution
(d) a feasible solution only at the origin
Show Answer & Explanation

Answer: (c) no feasible solution

Explanation:
1. From \(3x+5y\le15\) with \(x,y\ge0\) we get \(x+y\le\dfrac{3x+5y}3\le5\) (the largest value of \(x+y\) is \(5\) at \((5,0)\)).
2. So \(x+y\ge8\) cannot hold: the feasible region is empty.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 6: The number of corner points of the L.P.P. maximise \(Z=20x+3y\) subject to \(x+y\le5\), \(2x+3y\le12\), \(x,y\ge0\) is
(a) \(3\)
(b) \(4\)
(c) \(2\)
(d) \(1\)
Show Answer & Explanation

Answer: (b) \(4\)

Explanation:
1. The vertices are \((0,0)\), \((5,0)\), \((3,2)\) (from \(x+y=5\) and \(2x+3y=12\)) and \((0,4)\).
2. There are \(4\) corner points.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 7: The maximum value of \(z=3x+2y\) subject to \(x+2y\ge2\), \(x+2y\le8\), \(x,y\ge0\) is
(a) \(32\)
(b) \(40\)
(c) \(16\)
(d) \(24\)
Show Answer & Explanation

Answer: (d) \(24\)

Explanation:
1. The region is a strip between two parallel lines with vertices \((2,0)\), \((8,0)\), \((0,4)\), \((0,1)\).
2. \(z=6,\ 24,\ 8,\ 2\); the maximum is \(24\) at \((8,0)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 8: The maximum value of \(z=5x+3y\) subject to \(3x+5y\le15\), \(5x+2y\le10\), \(x,y\ge0\) is
(a) \(\dfrac{235}{19}\)
(b) \(\dfrac{325}{19}\)
(c) \(\dfrac{523}{19}\)
(d) \(\dfrac{532}{19}\)
Show Answer & Explanation

Answer: (a) \(\dfrac{235}{19}\)

Explanation:
1. The lines meet at \(\left(\dfrac{20}{19},\dfrac{45}{19}\right)\); the other vertices are \((0,0)\), \((2,0)\), \((0,3)\).
2. \(z=0,\ 10,\ 9,\ \dfrac{100+135}{19}=\dfrac{235}{19}\); the maximum is \(\dfrac{235}{19}\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 9: The constraints \(-x_1+x_2\le1\), \(-x_1+3x_2\le9\), \(x_1,x_2\ge0\) define
(a) a bounded feasible space
(b) an unbounded feasible space
(c) both bounded and unbounded feasible space
(d) no feasible space
Show Answer & Explanation

Answer: (b) an unbounded feasible space

Explanation:
1. The origin satisfies both inequalities, and \((t,0)\) satisfies them for every \(t\ge0\).
2. So the region extends indefinitely: it is unbounded.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 10: A brick manufacturer has depots \(A\) and \(B\) with stocks of \(30000\) and \(20000\) bricks, and orders of \(15000\), \(20000\), \(15000\) from builders \(P\), \(Q\), \(R\). Cost per \(1000\) bricks (Rs.): from \(A\) to \(P,Q,R\): \(40,20,20\); from \(B\) to \(P,Q,R\): \(20,60,40\). If \(x\) and \(y\) thousand bricks go from \(A\) to \(P\) and \(Q\), the L.P.P. to minimise cost is
(a) Minimise \(Z=40x-20y\) subject to \(x+y\ge15\), \(x+y\le30\), \(x\ge15\), \(y\le20\), \(x,y\ge0\)
(b) Minimise \(Z=40x-20y\) subject to \(x+y\ge15\), \(x+y\le30\), \(x\le15\), \(y\ge20\), \(x,y\ge0\)
(c) Minimise \(Z=40x-20y\) subject to \(x+y\ge15\), \(x+y\le30\), \(x\le15\), \(y\le20\), \(x,y\ge0\)
(d) Minimise \(Z=40x-20y\) subject to \(x+y\ge15\), \(x+y\le30\), \(x\ge15\), \(y\ge20\), \(x,y\ge0\)
Show Answer & Explanation

Answer: (c) Minimise \(Z=40x-20y\) subject to \(x+y\ge15\), \(x+y\le30\), \(x\le15\), \(y\le20\), \(x,y\ge0\)

Explanation:
1. Depot \(A\) sends \(30-x-y\ge0\) to \(R\); depot \(B\) sends \(15-x\ge0\) to \(P\), \(20-y\ge0\) to \(Q\) and \(x+y-15\ge0\) to \(R\).
2. These give \(x+y\le30\), \(x\le15\), \(y\le20\), \(x+y\ge15\), and the cost simplifies to \(40x-20y+1500\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 11: The inequalities \(5x+4y\ge20\), \(x\le6\), \(y\le4\) together with \(x,y\ge0\) form
(a) a square
(b) a rhombus
(c) a triangle
(d) a quadrilateral
Show Answer & Explanation

Answer: (d) a quadrilateral

Explanation:
1. The line \(5x+4y=20\) meets the \(x\)-axis at \((4,0)\) and the line \(y=4\) at \(\left(\dfrac45,4\right)\).
2. The region has the vertices \((4,0)\), \((6,0)\), \((6,4)\), \(\left(\dfrac45,4\right)\): a quadrilateral that is neither a square nor a rhombus.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 12: Maximise \(Z=4x+6y\) subject to \(3x+2y\le12\), \(x+y\ge4\), \(x,y\ge0\). The maximum is
(a) \(36\) at \((0,6)\)
(b) \(16\) at \((4,0)\)
(c) \(24\) at \((0,4)\)
(d) \(24\) at \((6,0)\)
Show Answer & Explanation

Answer: (a) \(36\) at \((0,6)\)

Explanation:
1. The vertices are \((4,0)\), \((0,4)\) and \((0,6)\), giving \(Z=16\), \(24\) and \(36\).
2. The maximum is \(36\) at \((0,6)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 13: \(Z=7x+y\) is to be minimised subject to \(5x+y\ge5\), \(x+y\ge3\), \(x\ge0\), \(y\ge0\). The minimum value of \(Z\) occurs at
(a) \((3,0)\)
(b) \(\left(\dfrac12,\dfrac52\right)\)
(c) \((7,0)\)
(d) \((0,5)\)
Show Answer & Explanation

Answer: (d) \((0,5)\)

Explanation:
1. The corner points are \((0,5)\), \(\left(\dfrac12,\dfrac52\right)\) and \((3,0)\).
2. \(Z=5\), \(6\) and \(21\) respectively, so the minimum is at \((0,5)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 14: The true statement for the graphs of the inequalities \(3x+2y\le6\) and \(6x+4y\ge20\) is
(a) Both graphs are disjoint
(b) Both contain \((0,3)\)
(c) Both contain the point \((1,1)\)
(d) The region common to both is a triangle
Show Answer & Explanation

Answer: (a) Both graphs are disjoint

Explanation:
1. The second inequality is \(3x+2y\ge10\), while the first is \(3x+2y\le6\).
2. No point can satisfy both, so the graphs are disjoint.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 15: Shami wants to invest Rs. \(50000\) in savings certificates and PPF. She invests at least Rs. \(15000\) in savings certificates (amount \(x\)) and at least Rs. \(20000\) in PPF (amount \(y\)). Rates are \(8\%\) and \(9\%\) p.a. The L.P.P. to maximise yearly income is
(a) Maximise \(Z=0.08x+0.09y\) subject to \(x+y\le50000\), \(x\ge15000\), \(y\le20000\)
(b) Maximise \(Z=0.08x+0.09y\) subject to \(x+y\le50000\), \(x\le15000\), \(y\ge20000\)
(c) Maximise \(Z=0.08x+0.09y\) subject to \(x+y\le50000\), \(x\ge15000\), \(y\ge20000\)
(d) Maximise \(Z=0.08x+0.09y\) subject to \(x+y\le50000\), \(x\le15000\), \(y\le20000\)
Show Answer & Explanation

Answer: (c) Maximise \(Z=0.08x+0.09y\) subject to \(x+y\le50000\), \(x\ge15000\), \(y\ge20000\)

Explanation:
1. The total cannot exceed \(50000\): \(x+y\le50000\).
2. 'At least' gives \(x\ge15000\) and \(y\ge20000\); income is \(0.08x+0.09y\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 16: The maximum value of \(P=x+3y\) subject to \(2x+y\le20\), \(x+2y\le20\), \(x,y\ge0\) is
(a) \(30\)
(b) \(10\)
(c) \(60\)
(d) \(26\)
Show Answer & Explanation

Answer: (a) \(30\)

Explanation:
1. The vertices are \((0,0)\), \((10,0)\), \(\left(\dfrac{20}3,\dfrac{20}3\right)\), \((0,10)\).
2. \(P=0\), \(10\), \(\dfrac{80}3\), \(30\); the maximum is \(30\) at \((0,10)\).

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

Question 17: A printing company earns Rs. \(10\) on each copy of magazine \(A\) and Rs. \(15\) on each copy of \(B\). Machines I, II, III take \((2,3)\), \((5,2)\), \((2,6)\) hours per copy of \((A,B)\) and are available for \(36\), \(50\) and \(60\) hours per week. The number of constraints (including the non-negativity conditions) is
(a) \(3\)
(b) \(5\)
(c) \(4\)
(d) \(6\)
Show Answer & Explanation

Answer: (b) \(5\)

Explanation:
1. One time constraint for each machine: \(2x+3y\le36\), \(5x+2y\le50\), \(2x+6y\le60\).
2. Adding \(x\ge0\) and \(y\ge0\) gives \(5\) constraints.

Teacher's Note:
1. Find the corner points of the feasible region and evaluate the objective function at each one.
2. Check which constraints are binding at a vertex, and remember the non-negativity conditions.

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Linear Programming Objective Questions & Solutions for JEE Mathematics

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